1 / 31

The Uses of Irrationality

The Uses of Irrationality. John D Barrow. Paper Sizes. The Square Root of Two. Does 2 = P/Q with P,Q integers ??. 1. 2. 2 = 1.414213562..  99/70. 1. Assume 2 = P/Q and P,Q integers with no common divisor

nigel-west
Download Presentation

The Uses of Irrationality

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The Uses of Irrationality John D Barrow

  2. Paper Sizes

  3. The Square Root of Two Does 2 = P/Q with P,Q integers ?? 1 2 2 = 1.414213562..  99/70 1 Assume2 = P/Q and P,Q integers with no common divisor P2 = 2Q2 so P2 is even and P must be even as well (because even = even x even or even x odd) So P = 2N and Q2 = ½ x 4N2 = 2N2 So Q2 and Q are both even as well. Therefore P and Q have a common divisor 2. This contradicts our original hypothesis – which is therefore false. 2 cannot be written as a rational fraction, P/Q, with P,Q integers: it is called an ‘irrational’ number Euclid Book 10, but known to the Pythagoreans

  4. Nice Irrational Aspect Ratios rotate 2 r 1 1 r 2 Height: width = 2/r Height: width = r/1 r/1 = 2/r r2 = 2 and r = 2

  5. And so on….. If you cut format A(N) paper parallel to its shorter side into two equal pieces of paper, these will have format A(N+1) All sizes rounded to the nearest millimetre

  6. International Standard Paper Sizes

  7. Tolerances • ±1.5 mm (0.06 in) for dimensions up to 150 mm (5.9 in) • ±2 mm (0.08 in) for lengths in the range 150 to 600 mm (5.9 to 23.6 in) • ±3 mm (0.12 in) for any dimension above 600 mm (23.6 in)

  8. The Lichtenberg Ratio Georg Christoph Lichtenberg wrote to Johann Beckmann on 25th October 1786 about the advantages ofa 2 paper-size ratio 1742-99 ‘Love is blind but marriage restores its sight’

  9. A-series Paper Sizes

  10. B-series Paper Sizes Length and width of B(n) are the geometric mean size of A(n) and A(n-1): B(n) = [A(n) x A(n-1)] eg size of B1 is (A1 x A0) size Beware Japanese standard B paper sizes! Japanese A series has the usual 2 scaling but Japanese B series is defined by the arithmetic mean not the geometric mean. This introduces other magnification scalings and is not used internationally.

  11. B-series Paper Sizes

  12. The Deep Magic of Xerox Machines All A series paper enlargements and reductions are by factors of 2 = 1.41 = 141% for enlargements and 1/2 = 0.71 = 71% for reductions 71 %, 84%, 119%, 141% 1/2, 1/2. 2, 2 A3  A4, B4  A4, A4  B4, A4  A3 B5  A4, A5  A4

  13. Photos of Xerox Machine Control Panels ‘…looks just like his dad’

  14. Go Forth and Multiply 0.71 x 0.71 = 0.504, 0.71 x 0.504 = 0.3579, 0.3579 x 0.71 = 0.2544 1/0.71 = 1.408, 1.408/0.71 = 1.983, 1.983/0.71 = 2.793, 2.793/0.71 = 3.968

  15. Newspapers Broadsheet 29½ ” x 23½” (750 x 600 mm) -- depth x width) Tabloid (or ‘Compact’) 17” x 11” (430 x 280 mm) Berliner 18.5” x 12.5” (470 x 315 mm)

  16. C-series Paper Sizes C(n) = [A(n)xB(n)] C4 envelope A4 letter fits easily inside unfolded C5 envelope A4 letter fits easily inside Folded in half A4: 297 x 210 mm C6 envelope

  17. Uses

  18. Some Handy Formulae for Paper Tigers

  19. Quantum Gravitational Paper! A233 has an area 2-233 m2 (10-35 m)2  Gh/c3 = 1 Planck area unit Breakdown of classical and quantum picture of space! S = kB (surface area)/(Planck area) Bekenstein-Hawking Entropy

  20. Areas and Paper Weights • A0 has area 1 sq m • A4 has area 1/24 = 1/16 sq m • Common paper quality is 5 gm per page for A4 • C4 envelope weighs less than 20 gm • You can put 16 A4 pages in the envelope before it weighs • (16 x 5) + 20 = 100gm • Good for calculating weight of stacked papers

  21. Technical Drawing Pen Nibs Standard sizes: 2.00mm, 1.40mm, 1.00mm, 0.70mm, 0.50mm 0.35mm, 0.25 mm, 0.18mm, 0.13mm They all differ by a factor of approx 2 = 1.4.. Four colour-coded standards: 0.25 , 0.35 , 0.50, 0.70 mm Draw with 0.35 mm pen on A3 paper and reduce to A4 You can draw on the copy with a 0.25mm pen. Stencil templates have similar scaling 5mm high letters have thickness 0.5mm (brown nib) in A0 Copy to A1 and text is 3.5mm high and 0.35 mm thick (yellow nib)

  22. Real Irrationality: American Paper Sizes USA, Canada and Mexico are the only three major countries that don’t use the International standard A, B and C series paper sizes Letter” (216 × 279 mm), “Legal” (216 × 356 mm), “Executive” (190 × 254 mm), “Ledger/Tabloid” (279 × 432 mm) US photocopiers usually have two or more paper trays. Enlarging of a “Letter” page onto “Legal” paper will cut off margins! Some copiers offer the larger “Ledger” layout, but it also has a different aspect ratio and changes the margins during magnification or reduction. Hopelessly inefficient and inconvenient!

  23. The Golden Ratio /1 = 1/(-1) 2 -  -1 =0 • = ½ {1 + 5} = 1.6180339… 1/ =  - 1 = 0.6180339…

  24. Euclid’s Definition c 300 BC  1 A C B • = AC/CB = AB/AC = ( + 1)/  • 2 -  -1 =0 = The real number that is farthest from any rational number

  25. Two good approximations •  (5/6) and   7/5e Accurate to 1.2 x 10-5 and 1.6 x 10-5 And Continued fractions again… Rational approximations are 1, 1 + 1/1, 1 + 1/(1+1), etc ie 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, …  = Ie successive approximations are ratios of consecutive Fibonacci numbers! 1,1,2,3,5,8,13,…..

  26. Medieval Vellum and Paper Folding quarto Fold over In half again folio Fold over octavo If start with coloured side up: you always have Page 1 white Pages 2-3 coloured Pages 4-5 white Pages 6-7 coloured etc No matter how many times you fold Flesh side of vellum will always face flesh and hair will face hair

  27. Manuscript of Euclid’s Elements Adelard of Bath, 4th Dec 1480

  28. Gutenberg Bible

  29. Medieval Book Page Canons Margin proportions 2:3:4:6 (inner:top:outer:bottom) when the page proportion is 2:3 [more generally 1:R:2:2R for page proportion 1:R (Van der Graaf)] Height of text area = page width for R=3/2

  30. Tschichold’s Construction 1/9th page ht circle Divide into 1/9ths 2/9th of page ht 2/9th 1/9th of page width Type area height = page width 2:3 page size ratio = text size ratio Give 2:3:4:6 inner:top:outer:bottom margin ratios Page to text area ratio = (3/2)2 = 9/4

More Related